Finite Group Databasebeta

← #8387

Subgroups — #8387

Order 256.

499 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic
2#1800C211yesmaximal, characteristic, complemented
3#1546C212yesmaximal, complemented
4#1486C212yesmaximal, complemented
5#1460C211yesmaximal, characteristic, complemented
6C22×#41—21no—
7#547—24no—
8#547—24no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×#123—22no—
11C2×C22⋊Q8—21no—
12C2×#121C412yes—
13C2×C4×D4C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4C412yes—
16#431C2212yes—
17#423C2212yes—
18C2×#100—21no—
19#343—21no—
20#332—24no—
21D4⋊C8—24no—
22#323—22no—
23C2×#41—28no—
24C2×#41—44no—
25C2×#41—42no—
26C2×#41—42no—
27C22×Q8—21no—
28C22×D4—22no—
29C22×D4C2×C411yescharacteristic
30C22×D4D411yescharacteristic
31C23×C4—22no—
32C23×C4—21no—
33C23×C4C2×C412yes—
34C2×M16—22no—
35C4⋊Q8—22no—
36C4⋊Q8D414yes—
37#123—216no—
38C22⋊Q8—28no—
39#121—24no—
40#121—24no—
41C4×D4—22no—
42C4×D4—22no—
43C4×D4—22no—
44C4×D4D414yes—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—24no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4C2×C411yescharacteristic
50C2×C4⋊C4C2×C412yes—
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—22no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4D412yes—
55C2×C42C2311yescharacteristic
56C4⋊C8—22no—
57C4⋊C8—22no—
58C4⋊C8—22no—
59#100—42no—
60C22⋊C8—22no—
61C4×C8—22no—
62C24—21no—
63C24C22⋊C412yes—
64#41—416no—
65#41—416no—
66C2×Q8—24no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—42no—
70C2×D4—28no—
71C2×D4—21no—
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—42no—
75C2×D4—44no—
76C2×D4—44no—
77C2×D4C22⋊C412yes—
78C2×D4D812yes—
79C2×D4SD1612yes—
80C22×C4—28no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—24no—
86C22×C4—44no—
87C22×C4—44no—
88C22×C4—44no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—42no—
92C22×C4C22⋊C412yes—
93C22×C4C22×C411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96M16—42no—
97M16—42no—
98C2×C8—42no—
99C2×C8—42no—
100C4⋊C4—28no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—24no—
104C4⋊C4—22no—
105C4⋊C4C22⋊C412yes—
106C4⋊C4C22⋊C414yes—
107C22⋊C4—28no—
108C22⋊C4—24no—
109C22⋊C4—24no—
110C42C2×D412yes—
111C42C2×D412yes—
112C23—22no—
113C23—22no—
114C23—22no—
115C23—42no—
116C23—42no—
117C23—42no—
118C23—42no—
119C23—41no—
120C23—41no—
121C23#10214yes—
122C23C2×C22⋊C411yescharacteristic
123Q8—44no—
124Q8—44no—
125D4—24no—
126D4—24no—
127D4—24no—
128D4—42no—
129D4—416no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—416no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4C2×C22⋊C411yescharacteristic
146C2×C4C2×C22⋊C411yescharacteristic
147C2×C4#12012yes—
148C2×C4#12012yes—
149C2×C4C2×D811yescharacteristic
150C2×C4C2×SD1611yescharacteristic
151C2×C4AGL(1,ℤ/8)12yes—
152C8—82no—
153C8—82no—
154C22—21no—
155C22—21no—
156C22—24no—
157C22—44no—
158C22—44no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22—82no—
165C22—82no—
166C22#37911yescharacteristic
167C22C2×#10211yescharacteristic
168C22#41811yescharacteristic
169C4—22no—
170C4—21no—
171C4—42no—
172C4—42no—
173C4—44no—
174C4#41312yes—
175C4#44712yes—
176C4#45012yes—
177C2—21no—
178C2—21no—
179C2—44no—
180C2—82no—
181C2#178511yescharacteristic
182C2#178611yescharacteristic
183C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #1800C2semidirect
3. #1546C2semidirect
4. #1486C2semidirect
5. #1460C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5472
#18002
#15462
#14862
#14602